Space of modular forms of level 686, weight 2, and Character 686.e

• Label: 686.2.e
• Level: $686$
• Weight: $2$
• Character orbit: 686.e
• Rep. character: $\chi_{686}(99,\cdot)$
• Character field: $\Q(\zeta_{7})$
• Dimension: $132$
• Newform subspaces: $6$
• Sturm bound: $196$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 686 = 2 \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	686.e (of order \(7\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 49 \)
Character field:			\(\Q(\zeta_{7})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(196\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(686, [\chi])\).

	Total	New	Old
Modular forms	672	132	540
Cusp forms	504	132	372
Eisenstein series	168	0	168

Trace form

\( 132 q + 2 q^{3} - 22 q^{4} + 6 q^{5} - 8 q^{6} - 28 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(686, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
686.2.e.a	$686$	$2$	686.e	49.e	$7$	$18$	$3$	$5.478$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None					98.2.e.a	$2$	$0$	\(-3\)	\(-3\)	\(6\)	\(0\)	$7$	$\mathrm{SU}(2)[C_{7}]$	\(q-\beta _{13}q^{2}+(-\beta _{1}+\beta _{14})q^{3}+(-1+\cdots)q^{4}+\cdots\)
686.2.e.b	$686$	$2$	686.e	49.e	$7$	$18$	$3$	$5.478$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None					98.2.e.b	$2$	$0$	\(3\)	\(5\)	\(0\)	\(0\)	$7$	$\mathrm{SU}(2)[C_{7}]$	\(q-\beta _{8}q^{2}+\beta _{3}q^{3}+(-1+\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
686.2.e.c	$686$	$2$	686.e	49.e	$7$	$24$	$4$	$5.478$		None					98.2.g.b	$2$	$0$	\(-4\)	\(-7\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{7}]$
686.2.e.d	$686$	$2$	686.e	49.e	$7$	$24$	$4$	$5.478$		None					98.2.g.b	$2$	$0$	\(-4\)	\(7\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{7}]$
686.2.e.e	$686$	$2$	686.e	49.e	$7$	$24$	$4$	$5.478$		None					98.2.g.a	$2$	$0$	\(4\)	\(-7\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{7}]$
686.2.e.f	$686$	$2$	686.e	49.e	$7$	$24$	$4$	$5.478$		None					98.2.g.a	$2$	$0$	\(4\)	\(7\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{7}]$

Decomposition of \(S_{2}^{\mathrm{old}}(686, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(686, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(343, [\chi])\)\(^{\oplus 2}\)